What Is a Maclaurin Series?
A Maclaurin series is a Taylor series with its center pinned at zero, the expansion of a function as a polynomial in plain powers of x:
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ⋯
Every coefficient is a derivative of f evaluated at 0, divided by a factorial. Because the center is the origin, the terms are the simplest possible, xⁿ rather than (x−a)ⁿ, which is why Maclaurin expansions are the ones people memorize, tabulate, and use for quick estimates near zero. The name honors Colin Maclaurin, an 18th-century Scottish mathematician, though the series is a special case of Taylor's more general construction; conceptually there's exactly one idea here, viewed from its most convenient vantage point.
How to Use This Calculator
Enter a function and an order n. The calculator differentiates repeatedly, evaluates each derivative at 0, divides by the factorial, and assembles the polynomial, every coefficient's origin displayed as a step. The graph overlays the function (solid) with its Maclaurin polynomial (dashed) so the near-zero hugging and far-from-zero divergence are visible at a glance. For expansions around a nonzero center, use the general Taylor Series Calculator; this page is the specialized, streamlined a = 0 version.
The Five Expansions Worth Memorizing
| Function | Maclaurin series | Valid for |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + ⋯ | all x |
| sin x | x − x³/3! + x⁵/5! − ⋯ | all x |
| cos x | 1 − x²/2! + x⁴/4! − ⋯ | all x |
| 1/(1−x) | 1 + x + x² + x³ + ⋯ | |x| < 1 |
| ln(1+x) | x − x²/2 + x³/3 − ⋯ | −1 < x ≤ 1 |
These five generate countless others through substitution and combination: replace x with −x² in the eˣ series and the expansion of e^(−x²), the Gaussian, which has no elementary antiderivative, appears instantly, letting you integrate it term by term. That substitution trick is often faster than differentiating from scratch, and it's how professionals actually produce series in practice.
Worked Example
Expand cos(x) to order 6. The derivatives cycle, cos, −sin, −cos, sin, so at zero they cycle through 1, 0, −1, 0. Only even orders survive:
cos x ≈ 1 − x²/2! + x⁴/4! − x⁶/6! = 1 − x²/2 + x⁴/24 − x⁶/720
The all-even structure isn't an accident: cosine is an even function (cos(−x) = cos x), and even functions have only even powers in their expansions, just as odd functions like sine keep only odd powers. Symmetry halves the work before differentiation even begins.
Common Mistakes to Avoid
- Forgetting the factorial denominators. The x⁴ coefficient of cos is 1/24, not 1/4, each term divides by n!, and the factorials are what make the series converge so fast.
- Expecting accuracy far from zero. A Maclaurin polynomial is built entirely from information at the origin; approximating sin(6) with a few terms centered at 0 fails badly. Far-away points want a shifted Taylor center.
- Ignoring the radius of convergence. The geometric series 1/(1−x) diverges the instant |x| ≥ 1, no matter how many terms you take, check the validity column before substituting. The Radius of Convergence Calculator automates the check.
- Dropping alternating signs. The sign patterns in sin, cos, and ln(1+x) carry the cancellation that makes the approximations track the function, a lost minus sign wrecks everything downstream.
Real-World Applications
Maclaurin series are how machines compute the functions on their buttons: floating-point libraries evaluate sin, cos, exp, and ln with polynomial approximations descended directly from these expansions, engineered to be exact to the last bit of a double-precision number. Physics leans on the first terms constantly, sin θ ≈ θ (the pendulum), cos θ ≈ 1 − θ²/2 (small-angle energy), √(1+x) ≈ 1 + x/2 (relativistic corrections), each a Maclaurin truncation with a name and a job.
In probability and statistics, the moment generating function's Maclaurin coefficients are a distribution's moments, mean, variance, and beyond appear as derivatives at zero, and in signal processing, small-signal linearization of amplifier response is the first-order Maclaurin term with distortion measured by the higher ones. Wherever a system is studied near its resting state, the expansion at zero is the natural microscope.
Frequently Asked Questions
What's the difference between Taylor and Maclaurin series?
Center only: Taylor expands about any point a, Maclaurin fixes a = 0. Every Maclaurin series is a Taylor series; the separate name survives because the zero-centered case is so common it earned its own label.
Why do even functions get only even powers?
Substituting −x must reproduce f(x) for an even function, and (−x)ⁿ = xⁿ only for even n, odd-power coefficients are forced to zero. The mirror argument kills even powers for odd functions. The calculator's coefficient list shows the zeros explicitly.
How many terms do I need for a good approximation?
Depends on how far from zero you evaluate and the function's growth: for |x| ≤ 1, five or six terms of the everywhere-convergent series (eˣ, sin, cos) already give several correct decimals. The Taylor Remainder Calculator turns "how many terms" into a rigorous error bound.
Can I integrate or differentiate a Maclaurin series term by term?
Yes, inside its radius of convergence, differentiating the sin series term by term produces the cos series exactly. Term-by-term integration is how non-elementary integrals like ∫e^(−x²)dx get usable series answers, the basis of the error function in statistics.
What if my function is undefined at 0, like ln(x) or 1/x?
Then no Maclaurin series exists, the coefficients f(0), f'(0), … don't. The fixes: shift the center (Taylor at a = 1 for ln) or expand a shifted relative like ln(1+x). It's why tables list ln(1+x) rather than ln(x).
Why do the factorials appear in the coefficients at all?
They're what makes the matching work: differentiating xⁿ n times produces n!, so for the series' nth derivative at 0 to equal f's, the coefficient must pre-divide by n!. Each factorial is a bookkeeping correction ensuring polynomial and function agree derivative-for-derivative at the center, and, serendipitously, the same factorials drive the series' fast convergence.
How is a Maclaurin series related to a Taylor series?
A Maclaurin series is simply a Taylor series centered at zero. It is the most common expansion point because many standard functions have clean coefficients there, and shifting the center to any other point recovers the general Taylor series.